The distribution of SAT scores of all college-bound seniors taking the SAT in 2014 was approximately normal with a mean of $1497 $ and standard deviation of $ 322 $. Let $X$ represent the score of a randomly selected tester from this group. Find $P(1497<X<1819)$. You may round your answer to two decimal places.
Solution: Representing probability with area Since we know the distribution of SAT scores is normally distributed, the probability $P(1497<X<1819)$ can be found by calculating the shaded area between $X=1497$ and $X=1819$ in the corresponding normal distribution: $531$ $853$ $1175$ $1497$ $1819$ $2141$ $2463$ $ \mu_X = 1497$ $ \sigma_X = 322$ $ P(1497<X<1819)$ Calculating shaded area We can use the "normalcdf" function on most graphing calculators to find the shaded area: $\begin{aligned} &\text{normalcdf:} \\\\ &\text{lower bound: } 1497 \\\\ &\text{upper bound: } 1819 \\\\ &\mu=1497 \\\\ &\sigma=322 \end{aligned}$ Output: $\approx0.3413$ [Why do we use normalcdf instead of normalpdf?] Answer $P(1497<X<1819)\approx0.34$ [Is there another way?]